To make both expressions {(x^2 + y^2 – 1), (x^2 – y^2 – 1)} square

$x^2 \; + \; y^2 \; - \; 1$
$x^2 \; - \; y^2 \; - \; 1$

Here are the first few solutions

$9^2 \; + \; 8^2 \; - \; 1 \; = \; 12^2$
$9^2 \; - \; 8^2 \; - \; 1 \; = \; 4^2$

$129^2 \; + \; 64^2 \; - \; 1 \; = \; 144^2$
$129^2 \; - \; 64^2 \; - \; 1 \; = \; 112^2$

$649^2 \; + \; 216^2 \; - \; 1 \; = \; 684^2$
$649^2 \; - \; 216^2 \; - \; 1 \; = \; 612^2$

$2049^2 \; + \; 512^2 \; - \; 1 \; = \; 2112^2$
$2049^2 \; - \; 512^2 \; - \; 1 \; = \; 1984^2$

$5001^2 \; + \; 1000^2 \; - \; 1 \; = \; 5100^2$
$5001^2 \; - \; 1000^2 \; - \; 1 \; = \; 4900^2$

$10369^2 \; + \; 1728^2 \; - \; 1 \; = \; 10512^2$
$10369^2 \; - \; 1728^2 \; - \; 1 \; = \; 10224^2$

$19209^2 \; + \; 2744^2 \; - \; 1 \; = \; 19404^2$
$19209^2 \; - \; 2744^2 \; - \; 1 \; = \; 19012^2$

Determine the recursion formula